Problem 12 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$\frac{9}{5} \text { to } \frac{11}{5}$$

Step-by-Step Solution

Answer

The ratio is \( \frac{9}{11} \) in lowest terms.

Step 1: Write Ratios as Fractions

First, when given a ratio like \( \frac{9}{5} \) to \( \frac{11}{5} \), recognize that this is a comparison of two fractions. The ratio can be expressed as a single fraction by placing the first fraction over the second: \[ \frac{\frac{9}{5}}{\frac{11}{5}} \].

Step 2: Multiply by the Reciprocal

Simplify the complex fraction by multiplying by the reciprocal. Multiplying by the reciprocal of the second fraction \( \frac{11}{5} \) means multiplying by \( \frac{5}{11} \). Thus, the expression becomes: \[ \frac{9}{5} \times \frac{5}{11} \].

Step 3: Cancel Common Factors

When multiplying \( \frac{9}{5} \) by \( \frac{5}{11} \), notice that the 5s cancel each other out: \[ \frac{9 \times 5}{5 \times 11} = \frac{9}{11} \].

Step 4: Simplify the Fraction

Check to see if \( \frac{9}{11} \) is in its simplest form. Since 9 and 11 have no common factors other than 1, \( \frac{9}{11} \) is already in its lowest terms.

Key Concepts

Fractions

Fractions are a way to express numbers as parts of a whole. They consist of a numerator (the top number) and a denominator (the bottom number).

• The numerator represents how many parts we have.
• The denominator shows the total number of equal parts in the whole.

For example, the fraction \( \frac{9}{5} \) indicates 9 parts out of a total of 5 parts. Fractions are essential in expressing ratios because they help compare two quantities easily.
When dealing with ratios in fractional form, it's often necessary to express them as a single fraction to simplify the comparison. In our exercise, the ratio is given as \( \frac{9}{5} \) to \( \frac{11}{5} \), which we transform into a single, complex fraction \( \frac{\frac{9}{5}}{\frac{11}{5}} \). By doing so, we can then proceed to apply mathematical operations to simplify and better understand the relationship between the two quantities.

Simplification

Simplification in math refers to the process of reducing an expression to its simplest form. When simplifying fractions, the goal is to make them as simple as possible by eliminating common factors from the numerator and the denominator.

• Look for common factors in both the numerator and denominator.
• Divide both by their greatest common divisor (GCD).

In the case of the complex fraction \( \frac{\frac{9}{5}}{\frac{11}{5}} \), simplification involves multiplying by the reciprocal of the fraction in the denominator. This transforms the problem into \( \frac{9}{5} \times \frac{5}{11} \). As you can see, multiplying by the reciprocal cancels out the 5s, simplifying to \( \frac{9}{11} \).
Simplifying ensures that we express the ratio in such a way that it is easy to interpret and compare, usually leaving it in its lowest terms where no further divisions are possible.

Reciprocal

A reciprocal of a number or a fraction is what you multiply by to get a product of one. For fractions, the reciprocal simply involves flipping the numerator and the denominator.

• The reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).

The concept of reciprocals is particularly useful in division. Dividing by a fraction is the same as multiplying by its reciprocal. In other words, instead of dividing, you multiply by the second fraction's flipped form.
In simplifying our initial ratio, we use the reciprocal when transforming the complex fraction \( \frac{\frac{9}{5}}{\frac{11}{5}} \) into a multiplication problem: \( \frac{9}{5} \times \frac{5}{11} \). By using the reciprocal, we find it easier to eliminate the complex fraction form and directly simplify to reach our final answer \( \frac{9}{11} \).
Understanding how to use reciprocals can make manipulating and solving fraction-based problems much smoother and more intuitive.